There weren't any English (equitation) instructors near my home or my work (45 miles apart) so I started to look online. I found 2 or 3 places an hour or so from home. By a stroke of luck, I really liked the first trainer I tried out and I've now been with her for four months.
1.) Discipline. I've spent time in the past with dressage/western people who swore they could teach equitation, but would always end up screwing up my seat.
2.) Cost. I knew I couldn't afford high-end lessons if I was having to drive 50+ miles in a Wrangler to get there.
3.) Riding experience/show history. I would expect any trainer to have some solid A/AA show experience. The clincher for me was my trainer also competed in the same zone/region I did in IHSA. I knew her college coach and she knew mine.
4.) How the barn is run/how horses are treated. If the barn has practices I'm not comfortable with, I walk away.
5.) Availability of leasing/showing school horses. I have no plans to own a horse in the next couple years, so this was important. Also, I was interested in being able to show school horses, even if it meant extra cost if I chose not to lease.
6.) Last and most importantly: Safety. Are helmets required, do current students have a good, solid seat and look to be control of their horses? Or do they look like train wrecks waiting to happen? Does the instructor progress students accordingly, or are they jumping within a month of taking lessons?
The first time I met my instructor, it was after a schooling horse show her barn had put on. I was able to stand around and watch her students, her lesson horses, speak with a few people. Good vibes all around. Everyone was polite and friendly, students looked safe, no drama, and I noticed my future trainer was at the gate, making sure her kids were ready, tallying up scores, etc.